Chapter 20 Risk and Return

Learning Objectives

1. Explain business risk and financial risk.2. Understand how to measure the business risk and financial risk.3. Understand the risk and return relationship for individual securities and a portfolio of

securities.4. Understand the capital asset pricing model (CAPM).

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Riskand

Return

Riskand

Uncertainty

Business riskand

Financial risk

Expected returnAnd SD for

Individual shares

Risk andReturn forPortfolio

CAPM

1. Risk and Uncertainty

1.1 Investment appraisal faces the following problems:(a) all decisions are based on forecasts;(b) all forecasts are subject to uncertainty;(c) this uncertainty needs to be reflected in the financial evaluation.

1.2 The decision maker must distinguish between:(a) Risk – can be quantifiable and be applied to a situation where there are

several possible outcomes and, on the basis of past relevant experience, probabilities can be assigned to the various outcomes that could prevail.

(b) Uncertainty – is unquantifiable and can be applied to a situation where there are several possible outcomes but there is little past experience to enable the probability of the possible outcomes to be predicted.

1.3 In investment appraisal the areas of concern are therefore the accuracy of the estimates concerning:(a) Project life(b) Predicted cash flows and associated probabilities(c) Discount rate used

2. Business Risk and Financial Risk

2.1 Business risk

3.1.1 Business Risk and its Measurement

(a) Business risk – refers to the relative dispersion in the company’s expected earnings before interest and taxes (i.e. EBIT).

(b) Operating gearing is a measure of the extent to which a firm’s operating costs are fixed rather than variable as this affects the level of business risk in the firm. Operating gearing can be measured in a number of different ways, including:

1.Fixed costs

orVariables costs

2.Fixed costs

orTotal costs

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3.% change in EBIT (or PBIT)

or% change in turnover

4.Contribution

PBIT or EBIT

Contribution is sales minus variable cost of sales.

2.1.2 Firms with a high proportion of fixed costs in their cost structures are known as having high operating gearing. => Higher business risk

2.1.3 Example 1Two firms have the following cost structures:

Firm A Firm B$m $m

Sales 5.0 5.0Variable costs (3.0) (1.0)Fixed costs (1.0) (3.0)EBIT 1.0 1.0

What is the level of operating gearing in each and what would be the impact on each of a 10% increase in sales?

Solution:

Operating gearing can be calculated as follows:Firm A Firm B

Fixed costs/variable costs 1/3 = 0.33 3/1 = 3

Firm B carries a higher operating gearing because it has higher proportion of fixed costs.

Its operating earnings will therefore be more volume-sensitive:

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Firm A Firm A Firm B Firm B$m 10%

increase$m 10%

increaseSales 5.0 5.5 5.0 5.5Variable costs (3.0) (3.3) (1.0) (1.1)Fixed costs (1.0) (1.0) (3.0) (3.0)EBIT 1 1.2 1 1.4

Firm B has enjoyed an increase in EBIT of 40% whilst Firm A has had an increase of only 20%. In the same way a decrease in sales would bring about a greater fall in B’s earning than in A’s.

2.1.4 Example 2If a company were to automate its production line to replace the workers currently paid an hourly wage, what would be the expected effect on its operating gearing?

Solution:

Swapping variable costs for fixed would increase the level of operating gearing.

2.2 Financial risk

2.2.1 Financial Risk and its Measurement

(a) Financial risk – is a direct result of the company’s financing decision.(b) The greater the level of debt, the more financial risk (of reduced dividends

after the payment of debt interest) to the shareholder of the company, so the higher is their required return.

(c) Financial gearing measures the relationship between shareholders’ capital plus reserves and capital or borrowings or both.

1. Equity gearing =Preference share capital + long-term debt

Ordinary share capital + reserve

2.Total or capital gearing =

Preference share capital + long-term debtTotal long-term capital

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Total long-term capital = Shares + reserves + PS + Long term debt

3. Interest gearing =Debt interest

PBIT

Notes:(a) Since preference shares are treated as debt finance, preference dividends

are treated as debt interest in this ratio.(b) For comparison purposes, the same ratio must be used consistently.(c) Capital gearing is used more than equity gearing.(d) Interest gearing is an income statement measure rather than a statement of

financial position one. It considers the percentage of the operating profit absorbed by interest payments on borrowings and as a result measures the impact of gearing on profits. It is more normally seen in its inverse form as the interest cover ratio.

2.2.2 The ratios can be calculated on either book or market values of debt and equity. There are arguments in favour of both approaches:(a) Market values:

(i) are more relevant to the level of investment made(ii) represent the opportunity cost of the investment made(iii) are consistent with the way investors measure debt and equity.

(b) Book values:(i) are not subject to sudden change due to the market factors(ii) are readily available.

2.2.3 Example 3The following excerpt has been obtained from the financial statements of ABC Co.

Statement of financial position excerpt2010 2009$000 $000

Total assets less current liabilities 158 139Non-current liabilities5% secured loan notes 40 40

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118 99

Ordinary share capital (50c shares) 35 358% Preference shares ($1 shares) 25 25Share premium account 17 17Revaluation reserve 10 -Income statement 31 22

118 99

Income statement excerpt2010 2009

$000 $000 $000 $000Gross profit 52 45Interest 2 2Depreciation 9 9Sundry expenses 14 11

(25) (22)Net profit 27 23Taxation (10) (10)Net profit after taxation 17 13Dividends:Ordinary shares 6 5Preference shares 2 2

(8) (7)Retained profit 9 6

Total market values are/were as follows: 2010 2009Ordinary shares (per share) 204c 195cPreference shares (per share) 80c 102c5% loan notes (per $100 nominal value) $108 $116

Calculate:

(a) Equity gearing(b) Capital gearing(c) Interest gearing

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For ABC Co using both statement of financial position and market values.

Solution:

(a) Equity gearing

2010 2009Book values

= 69.9% = 87.8%

Market values

= 44.3% = 52.7%

Note: Since preference shares are treated as debt, equity gearing could also be described as the debt/equity ratio.

(b) Capital gearing

2010 2009Book values

= 41.1% = 46.8%

Market values

= 30.7% = 34.5%

(c) Interest gearing2010 2009

Interest gearing

= 13.8% = 16.0%

2.2.4 Impact of financial gearing – where two companies have the same level of variability in earnings, the company with the higher level of financial gearing will have increased variability of returns to shareholders.

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2.2.5 Example 4Calculate the impact on Firm C of a 10% fall in sales and comment on your results:

$000Sales 10Variable costs (2)Fixed costs (5)EBIT 3Interest (2)EAIBT 1

Solution:$000 10% decrease ($000)

Sales 10 9Variable costs (2) (1.8)Fixed costs (5) (5)EBIT 3 2.2Interest (2) (2)EAIBT 1 0.2

The impact of a 10% decrease in sales has reduced operating earnings by (3 – 2.2)/3 = 26.67%.

The increased volatility can be explained by the high operating gearing in C.

However, C also has debt interest obligations. This financial gearing has the effect of amplifying the variability of returns to shareholders. The 10% drop in sales has caused the overall return to fall by (1 – 0.2)/1 = 80%. The additional 53.33% variation over and above the change in operating earnings is due to the use of debt finance.

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3. Expected Return and Standard Deviation for Individual Share(Dec 12)

3.1 Expected Return and Standard Deviation

(a) The expected return is the weighted average of all possible outcomes, with the weightings based on the probability estimates.

The formula for calculating an expected return is:

Expected return = Where: p = the probability of an outcome

x = the value of an outcome

(b) The standard deviation gives a measure of the extent to which outcomes vary around the expected return, as set out in the following formula:

3.2 Example 5The forecast information related to A Ltd’s share is as follows:

Event Return Ri Probability Pi

Boom 20% 0.2Growth 5% 0.6Recession -10% 0.2

Required:

Calculate the expected return and standard deviation of A Ltd’s share.

Solution:

Expected return:

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Event Return Ri Probability Pi Ri x Pi

Boom 20% 0.2 4%Growth 5% 0.6 3%Recession -10% 0.2 -2%

Expected return 5%

Standard deviationEvent Return Ri Probability

Pi

Deviation

Boom 20% 0.2 15% 45Growth 5% 0.6 0% 0Recession -10% 0.2 -15% 45

Variance 90

SD 9.49%

Question 1(a) Rising Star Company is considering investing in Project A and is provided with

following information:

State of Economy Probability Project A ReturnGood 25% 18%

Average 50% 10%Bad 25% -5%

Required:

(i) Determine the range and standard deviation for Project A. (3 marks)(ii) What does standard deviation measure in statistical sense? (2 marks)(iii) What does standard deviation measure in financial sense? (2 marks)

(b) Rising Star Company is considering investing in project B for the past three years and is provided with the following information:

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Year Project B return (%)2006 18%2005 10%2004 -5%

Required:

(i) Determine the range and standard deviation for Project B. (3 marks)(ii) Are the range and standard deviation for Project B the same as those for Project

A in (a)? Explain the differences, if any. (3 marks)(iii) Should Rising Star use the range or the standard deviation for investment

decisions? Why? (2 marks)(HKIAAT Paper III Financial Management June 2007 Q3(a)&(b))

4. Risk and Return for Portfolio of Securities

4.1 Diversification

4.1.1 An investor, knowing that a particular investment was risky, could decide to reduce the overall risk faced, by acquiring a second share with a different risk profile and so obtain a smoother average return. Reducing the risk in this way is known as diversification.

For example, an investor is not confined to a pure investment in either Ace’s shares or Bravo’s shares. Another possibility is to buy a portfolio (投資組合), in other words, to split the fund between the two companies.

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4.2 Portfolio expected return and risk(Jun 10, Dec 12)

4.2.1 Two-asset Portfolio Expected Return and Standard Deviation

(a) The expected return from a two-asset portfolio is as follows.

Where: X = the proportion invested in shares A or B

= the expected returns

(b) The formula for the standard deviation of a two-asset portfolio:

Where: = portfolio standard deviation = variance of investment A = variance of investment B

Cov(A,B) = covariance of A and B

(c) Covariance – measures the extent to which the returns on two investments ‘co-vary’ or ‘co-move’. If the returns tend to go up together and go down together then the

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covariance will be a positive number. If the returns on one investment move in the opposite direction to

the returns on another, then these securities will exhibit negative covariance.

If there is no co-movement at all, that is, the returns are independent of each other, the covariance will be zero.

The covariance formula is:

Cov(A,B) =

(d) Correlation coefficient – it has the same properties as the covariance but it measures co-movement on a scale of – 1 to + 1 which makes comparisons easier. Correlation coefficient = + 1, it is the case of perfect positively

correlated return for the two shares. Correlation coefficient = – 1, it means that the returns on one share

are an exactly opposite way to the returns on another. Correlation coefficient = 0, the two shares have no relationship with

each other, we are unable to show a line representing the degree of co-movement.

The correlation coefficient formula is:

4.2.2 The following graphs illustrate the three cases of correlation coefficient:

(a) Perfect positive correlation

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(b) Perfect negative correlation

(c) Zero correlation coefficient

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4.2.3 Example 6Suppose we are trying to forecast the possible rates of return of two investments over the next year. We make predictions as follows:

Event Probability Returns from A Returns from BRecession 0.2 10% 6%Stable 0.5 14% 15%Expansion 0.3 20% 11%

First, calculate the expected return and standard deviation of each investment. This tells us the risk and return of each security if held in isolation.

Investment AEvent Prob.

(P)Returns(RA%) P x RA

Recession 0.2 10 2 -5 5.0Stable 0.5 14 7 -1 0.5Expansion 0.3 20 6 +5 7.5

=15% Variance = 13.0

3.6%

Investment BEvent Prob.

(P)Returns(RB%) P x RB

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Recession 0.2 6 1.2 -6 7.2Stable 0.5 15 7.5 +3 4.5Expansion 0.3 11 3.3 -1 0.3

=12% Variance = 12.0

3.46%

Consider now constructing a portfolio consisting of one-half of the total amount invested in investment A and one-half in investment B.

Portfolio return Rp = 0.5 × 15% + 0.5 × 12% = 13.5%

Covariance of A and BEvent P RA RB

( )

x

( )P

Rec. 0.2 10 6 15 12 -5 -6 +6

Stable 0.5 14 15 15 12 -1 +3 -1.5

Exp. 0.3 20 11 15 12 +5 -1 -1.5

Cov(A,B) +3

Portfolio standard deviation, = = 2.78%

The portfolio has an expected return which is equal to the weighted average of the two investment returns. Its risk, as measured by standard deviation, is lower than either of the two original investments.

Question 2Lydia Wang currently owns 500 shares of common stocks of PLC Properties Holdings Ltd, which are valued at $50,000 ($100 per share). The expected annual return on PLC Properties’ shares is 10% with a standard deviation of 30%. Suppose Lydia has additional

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$50,000 to invest. She could use the money to buy another 500 shares in PLC Properties, or she could use it to buy 400 shares in e-Life Sciences Ltd, whose shares are trading at $125 each. The expected return on e-Life Sciences is also 10% with a standard deviation of 30%. The correlation coefficient between the returns on PLC and e-Life shares is 0.

Required:

(a) Should Lydia invest the additional $50,000 in 500 PLC shares or 400 e-Life shares? Why? (7 marks)

(b) Under what circumstances would Lydia be indifferent between the choice of buying 500 PLC shares or 400 e-Life shares? Explain. (Hint: Think of the necessary condition for portfolio diversification benefits). (3 marks)

(HKIAAT Paper III Financial Management June 2004 Q3(a))

Question 3Mr Gong currently has $2 million invested in a portfolio of corporate bonds. The bond portfolio has an expected annual return of 8% and an annual standard deviation of 12%. His daughter, Lily, has just finished her MBA degree and recommends her father consider investing half of the $2 million in a stock market index fund (such as the Hong Kong Tracker Fund) and remainder in the bond portfolio. The stock market index has an expected annual return of 14% with a standard deviation of 16%. The correlation between the index fund and the bond portfolio is 0.1.

Required:

(a) If Mr Gong follows his daughter’s advice, can he improve his expected return without taking on additional risk? Justify your answer. (6 marks)

(b) How does systematic risk differ from total risk? (4 marks)(HKIAAT Paper III Financial Management December 2004 Q3(a))

Question 4Suppose your uncle has left you $150,000 cash and 1,000 shares in CBSH currently worth $150,000. Unfortunately his will requires that the CBSH shares not to be sold for one year and the $150,000 cash must be entirely invested in one of the following two stocks: KHS Property or China Coal Mines. The table below summaries the correlation coefficients, expected annual returns, and annual standard deviations of the three stocks.

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Correlation coefficients Expected StandardCBSH KHS China Coal Return Deviation

CBSH 1 0.7 0.2 15% 20%KHS - 1 0.4 20% 30%China Coal Mines - - 1 25% 40%

Required:

(a) What is the portfolio that will deliver the highest attainable expected return under these restrictions? Show your workings. (5 marks)

(b) What is the portfolio that will have the lowest attainable risk under these restrictions? Show your workings. (5 marks)

(HKIAAT Paper III Financial Management December 2005 Q3(a))

Question 5Security A has an expected return of 10%, a standard deviation of expected returns of 30%, and a beta of 0.5. Security B has an expected return of 10%, a standard deviation of expected returns of 15%, and a beta of 1.2. The correlation coefficient between the two securities is 0.6.

Required:

(a) What is the expected return and standard deviation of a portfolio with 30% of the wealth invested in Security A and the remaining 70% invested in Security B?

(5 marks)(b) Suppose currently you hold no stocks and plan to buy and hold either Security A or

Security B alone. Which security would you choose? Why? (2 marks)(c) Now suppose you already hold a well-diversified portfolio and are considering adding

either Security A or Security B to your portfolio. Which security would you choose? Why? (3 marks)

(HKIAAT Paper III Financial Management June 2006 Q4(a))

Question 6In investment, fund managers try to increase the returns and decrease the variations in returns. A portfolio consists of 40% investment in stock M and 60% investment in stock N.

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Required:

(a) What is the name of the measure used to describe variations in returns? (2 marks)(b) If a stock M has returns of 2%, -12%, 27%, 22% and 18% in the past 5 years.

Calculate the arithmetic average return and the geometric average return. (4 marks)(c) Calculate the measure you suggest in part (a) for the stock mentioned in part (b).

Briefly explain why fund managers would try to minimize this measure. (5 marks)(d) If another stock N has the measure used to describe its variation in returns as

mentioned in part (a) equal to 5% and the correlation coefficient with stock M is 0.3, calculate the overall variation measure as described in part (a) for that portfolio.

(6 marks)(e) Suggest one way that an investor can use to reduce the overall variation in returns for

that portfolio which still consists of stock M and stock N. (3 marks)(HKIAAT Paper II Management Accounting and Finance December 2012 Q4)

4.3 Efficient frontier (or opportunity set, or feasible set)(Jun 10)

4.3.1 Example 7 Suppose an individual is able to invest in shares of Augustus, in shares of

Brown or in a portfolio made up from Augustus and Brown. Augustus is an ice cream manufacturer and so does well if the weather is

warm. Brown is an umbrella manufacturer and so does well if it rains. Because the weather is so changeable from year to year an investment in one

of these firms alone is likely to be volatile, whereas a portfolio will probably reduce the variability of returns.

Returns on shares in Augustus and BrownEvent Probability Returns on

AugustusReturns on

BrownWarm 0.2 20% -10%Average 0.6 15% 22%Wet 0.2 10% 44%Expected return 15% 20%

Standard deviation for Augustus and BrownProb. Returns on

AugustusReturns on

Brown

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0.2 20 5 -10 180.00.6 15 0 22 2.40.2 10 5 44 115.2

Variance 10 Variance 297.6

S.D. 3.162 S.D. 17.25

CovarianceP RA RB

( ) x

( )P

0.2 20 -10 15 20 5 -30 -30

0.6 15 22 15 20 0 2 0

0.2 10 44 15 20 -5 24 -24

Cov(A,B) -54

Risk-return correlations: Two-asset portfolios for Augustus and Brown

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Notes: The graph above shows the risk-return profile for alternative portfolios.

Portfolio K is very close to the minimum risk combination that actually occurs with a portfolio consisting of 84.6% in Augustus and 15.4% in Brown.

Take L and J as examples – they have almost the same risk levels but portfolio L dominates portfolio J because it has a better return. All the portfolios between K and A are inefficient because for each possibility there is an alternative combination of Augustus and Brown on the solid line K to B which provides a higher return for the same risk.

An efficient portfolio is a combination of investments which maximises the expected return for a given standard deviation.

Question 7Consider the expected returns and standard deviations of the following portfolios:

PortfolioA B C D E F G

Expected return (%) 10 12 15 18 19 20 20

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Standard deviation (%) 22 20 22 25 25 30 35

Required:

(a) Plot the portfolios on a graph. (5 marks)(b) Four of these portfolios are efficient, and three are not. Which are the inefficient ones?

(3 marks)(c) Suppose you are prepared to tolerate a standard deviation of 22%. Which portfolio

would you choose if you cannot borrow or lend? Why? (3 marks)(d) Suppose you can borrow and lend at the risk-free rate of 10%. What is your optimal

investment strategy if you are prepared to tolerate a standard deviation of 22%? What is the expected return associated with your optimal strategy? [Hint: The optimal strategy should consist of the risk-free asset and the optimal portfolio of risky assets.]

(9 marks)(Total 20 marks)

(HKIAAT Paper III Financial Management June 2005 Q2)

Question 8Lemon Company Limited utilizes excess cash to invest in securities, options including bonds and stocks. The expected returns of bonds and stocks are respectively 5% and 7% and the variances are 3% and 4%. The covariance is 2. 40% of investment is in bonds and the remainder is in stocks.

Required:

(a) Calculate the expected return and standard deviation of the portfolio. (6 marks)(b) What is the correlation coefficient between bonds and stocks? (4 marks)(c) Sketch a graph showing FOUR cases involving the variation of return with risk when

the correlation coefficient between bonds and stocks is 1, 0.5, -0.5 and -1 respectively. Explain your graph in brief. (6 marks)

(d) What is the conclusion you draw from the answer to part (c) in terms of portfolio management? (2 marks)

(e) Explain why it is unwise, from a portfolio management perspective, to invest only in stocks from the banking and property sectors in Hong Kong. (2 marks)

(Total 20 marks)(HKIAAT PBE Paper II Management Accounting and Finance June 2010 Q3)

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5. Capital Asset Pricing Model (CAPM)

5.1 Introduction

5.1.1 CAPM is developed based on portfolio theory. Portfolio theory considers the total risk and return of portfolios, while CAPM uses the systematic risk of individual securities.

5.2 Systematic and unsystematic risk(Dec 09, Jun 15)

5.2.1 Systematic Risk and Unsystematic Risk

If a particular security is the only investment you hold, you are subject to that security’s total risk. Total risk is the sum of:(a) Systematic risk (or market risk) – it is the risk of market wide factors

such as the state of economy. It will affect all companies in the same way (although to varying degrees), and it cannot be diversified away.

(b) Non-systematic (or unsystematic or business or unique) risk – it is the risk factors will impact each firm differently, depending on their circumstances. Diversification can almost eliminate unsystematic risk.

5.2.2 Exercise 1The following factors have impacted the volatility of the earnings of ABC Co, a manufacturer of chocolate biscuits and cereals:(a) increase in interest rates(b) increase in the price of cocoa beans

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(c) legislation changing the rules on tax relief for investments in non-current assets

(d) growth in the economy of the country where ABC Co is based(e) government advice on the importance of eating breakfast(f) industrial unrest in ABC Co’s main factory.

Are they examples of systematic or unsystematic risk?

Factors Type of risk(a) increase in interest rates S(b) increase in the price of cocoa beans U(c) legislation changing the rules on tax relief for investments

in non-current assets S

(d) growth in the economy of the country where ABC Co is based S

(e) government advice on the importance of eating breakfast U(f) industrial unrest in ABC Co’s main factory. U

Question 9In diversification, one can diversify one type of risk but fail to diversify another. Identify these risks and distinguish them with examples. (4 marks)

(HKIAAT PBE Paper II Management Accounting and Finance December 2009 Q6(a))

5.3 The security market line (SML)(Dec 10, Jun 15)

5.3.1 In the CAPM the relationship between risk as measured by beta and expected return is shown by the security market line.

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5.3.2 From the above graph, for example, shares perfectly correlated with the market return (M) will have a beta of 1.0 and are expected to produce an annual return of 11% in the circumstances of a risk-free rate of return at 6% and the risk premium on the market portfolio of shares over safe securities at 5% (point M). Shares which are twice as risky, with a beta of 2.0, will have an expected return of 16%; shares which vary half as much as the market index are expected to produce a return of 8.5% in this particular hypothetical risk-return line.

5.3.3 According to the CAPM all securities lie on the security market line, their exact position being determined by their beta. But what about shares J and I in the above graph? These are shares that are not in equilibrium.(a) Share J offers a particularly high level of return for the risk its holders to

bear. This will not last for long in an efficient market because investors are constantly on the prowl (四處尋覓) for shares like this. As they start to buy in large quantities the price will rise and correspondingly the expected return will fall. This will continue until the share return is brought on to the SML.

(b) Share I will be sold until the price falls sufficiently to bring about equilibrium, that is, I is placed on the SML.

5.3.4 The SML is often referred to in the form of an equation:

Rj = Rf + β × (Rm – Rf)

Where, Rf = risk free rate => government bond as the reference

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Rj = required rate of return on investment jRm = return on the market portfolio, E.g. HKSI, DJβ = index of systematic risk for security jRm – Rf = Risk premium

Question 10The expected rate of return on the market portfolio is 12%, and the risk-free rate is 6%. The expected return for the stock of Alpha Company is 16% using the capital asset pricing model (CAPM).

Required:(a) Determine the beta value for Alpha Company. (5 marks)(b) If you believe Alpha Company will be worth $100 per share one year from today, how

much are you willing to pay for one share today? => Refer to Chapter 25 (5 marks)(HKIAAT Paper III Financial Management December 2006 Q5(a))

5.4 Calculating beta (β)(Dec 10, Jun 15)

5.4.1 Beta

Beta is how much more or less volatile is a particular share going to be relative to the market. In other words, it shows the relationship between the particular share return with the market return.

Different sources use different references for market return. For example, some use the Hang Seng Index and some use MSCI Capital Market Index for HK Stock. In addition, different sources may use different time measurements such as weekly, monthly or yearly in measuring the returns. These all give different beta values. (MSCI = Morgan Stanley and Capital Group International)

The basic features of beta are:

β = 1, a 1% change in the market index return generally leads to a 1% change in the return on a specific share.

0 < β < 1, a 1% change in the market index return generally leads to a less than 1% change in the returns on a specific share.

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β > 1, a 1% change in the market index return generally leads to a greater return than 1% change in the returns on a specific share.

5.4.2 The slope of the regression line, beta, represents the relationship between the systematic risk of the individual investment and the overall market risk. It is therefore measured as follows:

Where:= Covariance of returns of security j and the market

= correlation coefficient between the security and the market= standard deviation of security j return= standard deviation of market return

5.4.3 Example 9ABC Co wishes to estimate its equity beta and has recorded the following information.

Month Return on the market portfolio (Rm)

Return on ABC’s equity (Rj)

1 2 3

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2 -1 -23 3 44 0 15 2 2.5

The following have been calculated for this data:Standard deviations:Market σm = ABC Coσj = Correlation coefficient ρm,j = +0.96

Solution:

Beta (β) =

If the risk-free rate of interest for some future period was anticipated to be 10% pa and the return on the market portfolio for the same period was expected to be 14%, then the required rate of return on ABC’s shares, according to CAPM, for the same period would be:Rj = Rf + β × (Rm – Rf) = 10% + 1.36 × (14% – 10%) = 15.44%

Question 11You have been analyzing two stocks and have assessed their characteristics as follows:

Stock Expected return Standard deviation BetaOne-i-Dea Ltd 12% 25% -0.5Sure-Bet Corporation 12% 10% 1.15

Required:

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(a) How do you create a portfolio consisting of the two stocks that will have a beta of 1.0?(4 marks)

(b) If you are a risk-averse investor, which stock will you choose? Why? (6 marks)(HKIAAT Paper III Financial Management June 2003 Q6(a))

5.5 Portfolio beta

5.5.1 While portfolio variance is not a weighted average of the individual asset variance, portfolio betas are a weighted average of the individual asset betas in the following equation:

= Weighted average of the betas of the assets in the portfolio = % of portfolio invested in asset i (weights)

5.5.2 Example 10Consider the given information for the following four securities, what is the portfolio beta?

Security Weight BetaA 0.133 4.03B 0.2 0.84C 0.267 1.05D 0.4 0.59

5.6 Assumptions of CAPM

5.6.1 The CAPM shows how the theoretical required return on a quoted security depends on its systematic risk. For simplicity, various assumptions are made:(a) a perfect capital market => all investors have the same info.(b) unrestricted borrowing or lending at the risk-free rate of interest(c) uniformity of investor expectations(d) all forecasts are made in the context of one time period only(e) Assume the investors are well-diversified.

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5.7 Advantages of CAPM

5.7.1 Advantages:(a) It provides a market based relationship between risk and return, and assessment

of security risk and rates of return given that risk.(b) It shows why only systematic risk is important in this relationship.(c) It is one of the best methods of estimating a quoted company’s cost of capital.(d) It provides a basis for establishing risk-adjusted discount rates for capital

investment projects.

5.8 Limitations of CAPM

5.8.1 Problems of CAPM include unrealistic assumptions and the required estimates being difficult to make.(a) The need to determine the excess return (Rm - Rf). Expected, rather than

historical, returns should be used, although historical returns are often used in practice.

(b) The need to determine the risk-free rate. A risk-free investment might be a government security. However, interest rates vary with the term of the lending.

(c) Errors in the statistical analysis used to calculate beta values. Betas may also change over time.

(d) The CAPM is also unable to forecast accurately returns for companies with low price/earnings ratios and to take account of seasonal “month-of-the-year” effects and “day-of-the-week” effects that appear to influence returns on shares.

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